battery dead after charging all night droid incerdibleSupport

I new to this form and just purchased new incredible and I let it go below 10% battery life and then charge. But in the morning after I unplug the charger in about 30 min I have to plug it into my car charger. The battery
shows to be charging when pluged. Does the the phone stop charging when level is reached and then somthing is killing the battery over night. Is any one else ever had this problem. If so please help I cant go like this.